Central Limit Theorem

Central Limit Theorem The central limit theorem states when the sample n is sufficiently large enough, then the distribution of the sample means 𝑥 approximates the normal model.

Conditions for CLT Sample can not be greater than 10% of the population 𝑛∗𝑝>10 𝑛∗𝑞>10 Samples must be independent Samples must be random

Old and new “AP” terms Parameter—characteristic of a population Statistic—characteristic of a sample 𝑝 𝑝−ℎ𝑎𝑡 is the observed mean of the sample(s). 𝑝 is our estimate of what the population parameter is. After we do all our samples, we say 𝑝 = 𝑥 Be certain to distinguish between parameter and statistic on the AP Exam.

AP TERm--Sampling distribution The sampling distribution of sample means is a distribution using the means from all possible samples of a specific size from a population. Be certain not to say “distribution” on the AP Exam if it is a “sampling distribution”.

Ap term—variability of a statistic The variability of a statistic (referred to as sampling error) is described by the spread of its sampling distribution. Larger samples give smaller spreads. Sampling error—the difference between the sample measure and the corresponding population measure due to the variation in the different samples.

Formulas—standard deviation of 𝒑 (observed value of our sample proportion) For a sample SD( 𝒑 ) =𝜎 𝑝 = 𝑝∗𝑞 𝑛 = 𝑝∗𝑞∗𝑛 𝑥 𝑜𝑟 𝑝 = 𝜇 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 For a sample mean 𝜎 𝑥 = 𝜎 𝑛

Nielson ratings states the mean hours spent watching tv by adolescents is 25 hours per week with a SD of 3 hours. If 20 adolescents are sampled, What is the chance the mean will be more than 26.3 hours per week? 𝜎 𝑝 = 3 20 = .671 𝑧= 26.3−25 .671 =1.9374 Normcdf(1.9374, 99)=2.26%

The average age of a vehicle today is 96 months with a SD of 16 months The average age of a vehicle today is 96 months with a SD of 16 months. If 36 cars are chosen at random, what is the chance they are between 90 and 100 months 𝜎 𝑝 = 16 36 =2.67 𝑧 𝑙𝑜𝑤𝑒𝑟_𝑏𝑜𝑢𝑛𝑑 = 90−96 2.67 =−2.25 & 𝑧 𝑢𝑝𝑝𝑒𝑟_𝑏𝑜𝑢𝑛𝑑 = 100−96 2.67 =1.5 Normal_cdf(-2.25,1.5) = 92.1% chance of success.

A short video https://www.youtube.com/watch?v=zr-97MVZYb0